On the $A_α$-spectra of trees

Let $G$ be a graph with adjacency matrix $A(G)$ and let $D(G)$ be the diagonal matrix of the degrees of $G$. For every real $α\in\left[ 0,1\right],$ define the matrix $A_α\left(G\right)$ as $A_α\left(G\right) =αD\left(G\right) +(1-α)A\left(G\right)$ where $0\leqα\leq1$. This paper gives several results about the $A_α$-matrices of trees. In particular, it is shown that if $T_Δ$ is a tree of maximal degree $Δ,$ then the spectral radius of $A_α(T_Δ)$ satisfies the tight inequality $ρ(A_α(T_Δ))<αΔ+2(1-α)\sqrt{Δ-1}.$ This bound extends previous bounds of Godsil, Lovász, and Stevanović. The proof is based on some new results about the $A_α$-matrices of Bethe trees and generalized Bethe trees. In addition, several bounds on the spectral radius of $A_α$ of general graphs are proved, implying tight bounds for paths and Bethe trees.

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